1986 AMC 12 第 10 题

Question

将 AHSME 的

120

个字母排列按字典顺序排列，把每个排列看成一个普通的五字母单词。这个表中第

86

个单词的最后一个字母是

The

120

permutations of AHSME are arranged in dictionary order, as if each were an ordinary five-letter word. The last letter of the

86

th word in this list is

A

H

S

M

E

答案：E

知识点：permutation lexicographic order counting

难度评级：1590

小提示：

按照首字母，将这些单词分为每组

4!

个的若干组

Group the words into blocks of

4!

according to their first letter

大提示：

在以

M

开头的一组中，再按第二个字母分组，只列出所需的那个小组

Within the

M

-block, group by the second letter and then list only the needed small block

解答：

每个首字母对应一组

4!=24

个单词。第

73

至第

96

个单词以

M

开头，所以第

86

个单词是这一组中的第

14

个。前

6

个以

MA

开头，接下来

6

个以

ME

开头，第

13

至第

18

个以

MH

开头。其中前两个是

MHAES

和

MHASE

。因此，第

86

个单词以 E 结尾。

所以正确答案是 E。

Each first letter occupies a block of

4!=24

words. Positions

73

through

96

begin with

M,

so the

86

th word is the

14

th word in that block. The first

6

begin with

MA,

the next

6

with

ME,

and the

13

th through

18

th begin with

MH.

The first two of those are

MHAES

and

MHASE.

Thus the

86

th word ends in E.

Therefore the correct answer is E.

1986 AMC 12 第 10 题

答案：E

难度评级：1590

小提示：

大提示：

解答：

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